Non-intrusive reduced order modelling of multi-fluid flows in oil reservoirs with uncertain rock properties

A novel variable-material non-intrusive reduced order model (NIROM) based on a Smolyak sparse grid interpolation method, a radial basis function (RBF) method and proper orthogonal decomposition (POD) has been developed for oil reservoirs with uncertain rock properties. This novel NIROM is constructed by using a two level interpolation method. The first level interpolation is constructed for the rock properties, and the second level is for the fluid dynamics. The NIROM is independent of governing equations, therefore, this method is easy to implement and easy to be extended for other applications as it does not require modifications to the source code. The novelty of this work is the use of the presented Smolyak gird and RBF interpolation based NIROM for oil reservoirs with uncertain rock properties. Another novelty is the use of Smolyak sparse grid to reduce the NR realisations of the reservoir simulator for constructing the NIROM and reduce the computational effort involved in creating the hyperspace of basis functions and coefficients from the snapshots of all the different realisations. The capability of this new NIROM has been numerically illustrated in two multiphase flows in porous media: a reservoir with eight baffles case and a 3D fluvial channel case with 22 uncertainties. By comparing the results of the novel NIROM against the solutions obtained from the high fidelity full model, it is shown that this model can result in a large reduction in the CPU cost (by a factor of about three orders) while much of the details of multiphase flow in porous media are captured. ∗Corresponding author: dh.xiao@imperial.ac.uk

[1]  R. H. Brooks,et al.  Hydraulic Properties of Porous Media and Their Relation to Drainage Design , 1964 .

[2]  Peter A. Forsyth,et al.  A Control Volume Finite Element Approach to NAPL Groundwater Contamination , 1991, SIAM J. Sci. Comput..

[3]  L. Durlofsky.  A triangle based mixed finite element–finite volume technique for modeling two phase flow through porous media , 1993 .

[4]  L. Durlofsky.  Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities , 1994 .

[5]  A. Megretski,et al.  Model Reduction for Large-Scale Linear Applications , 2003 .

[6]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[7]  Jan Dirk Jansen,et al.  Generation of Low-Order Reservoir Models Using System-Theoretical Concepts , 2004 .

[8]  S. Geiger,et al.  Combining finite element and finite volume methods for efficient multiphase flow simulations in highly heterogeneous and structurally complex geologic media , 2004 .

[9]  Gabriel Dimitriu,et al.  Comparative Study with Data Assimilation Experiments Using Proper Orthogonal Decomposition Method , 2009, LSSC.

[10]  Razvan Stefanescu,et al.  Numerical Simulations with Data Assimilation Using an Adaptive POD Procedure , 2009, LSSC.

[11]  M. A. Cardoso.  Reduced-Order Models for Reservoir Simulation , 2009 .

[12]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[13]  M. A. Cardoso,et al.  Use of Reduced-Order Modeling Procedures for Production Optimization , 2010 .

[14]  C. Farhat,et al.  Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations , 2011 .

[15]  J. Hahn,et al.  State-preserving nonlinear model reduction procedure , 2011 .

[16]  Han Chen,et al.  Blackbox Stencil Interpolation Method for Model Reduction , 2012 .

[17]  Feriedoun Sabetghadam,et al.  α Regularization of the POD-Galerkin dynamical systems of the Kuramoto-Sivashinsky equation , 2012, Appl. Math. Comput..

[18]  Hector Klie.  Unlocking Fast Reservoir Predictions via Nonintrusive Reduced-Order Models , 2013, ANSS 2013.

[19]  Charbel Farhat,et al.  The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows , 2012, J. Comput. Phys..

[20]  Caroline Sainvitu,et al.  Adaptive sampling strategies for non‐intrusive POD‐based surrogates , 2013 .

[21]  K. Morgan,et al.  Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions , 2013 .

[22]  Zeid M. Alghareeb,et al.  Optimum Decision-Making in Reservoir Managment Using Reduced-order Models , 2013 .

[23]  Ionel M. Navon,et al.  Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair , 2013 .

[24]  Mehdi Ghommem,et al.  Global-Local Nonlinear Model Reduction for Flows in Heterogeneous Porous Media Dedicated to Mary Wheeler on the occasion of her 75-th birthday anniversary , 2014, 1407.0782.

[25]  Juan Du,et al.  Non-linear model reduction for the Navier-Stokes equations using residual DEIM method , 2014, J. Comput. Phys..

[26]  M. Winter,et al.  Reduced-Order Modeling of Unsteady Aerodynamic Loads using Radial Basis Function Neural Networks , 2014 .

[27]  B. R. Noack,et al.  On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body , 2013, Journal of Fluid Mechanics.

[28]  S. Weiland,et al.  Tensor-based reduced order modeling in reservoir engineering: An application to production optimization ? , 2015 .

[29]  Ionel M. Navon,et al.  Non-intrusive reduced order modelling of the Navier-Stokes equations , 2015 .

[30]  C. Pain,et al.  Non‐intrusive reduced‐order modelling of the Navier–Stokes equations based on RBF interpolation , 2015 .

[31]  C. Lacor,et al.  A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition , 2015 .

[32]  M. Blunt,et al.  Reservoir Modeling for Flow Simulation by Use of Surfaces, Adaptive Unstructured Meshes, and an Overlapping-Control-Volume Finite-Element Method , 2015 .

[33]  B. R. Noack,et al.  On long-term boundedness of Galerkin models , 2013, Journal of Fluid Mechanics.

[34]  P. Breitkopf,et al.  A manifold learning-based reduced order model for springback shape characterization and optimization in sheet metal forming , 2015 .

[35]  L. Heltai,et al.  Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils , 2015 .

[36]  Stéphane Bordas,et al.  A fast, certified and "tuning free" two-field reduced basis method for the metamodelling of affinely-parametrised elasticity problems , 2016 .

[37]  Christopher C. Pain,et al.  Non-intrusive reduced order modelling of fluid–structure interactions , 2016 .

[38]  K. C. Hoang,et al.  An hp-proper orthogonal decomposition–moving least squares approach for molecular dynamics simulation , 2016 .

[39]  J. Percival,et al.  A balanced-force control volume finite element method for interfacial flows with surface tension using adaptive anisotropic unstructured meshes , 2016 .

[40]  K. Willcox,et al.  Data-driven operator inference for nonintrusive projection-based model reduction , 2016 .

[41]  Ionel M. Navon,et al.  2D Burgers equation with large Reynolds number using POD/DEIM and calibration , 2016 .

[42]  P. Salinas,et al.  Reservoir Modelling Using Parametric Surfaces and Dynamically Adaptive Fully Unstructured Grids , 2016 .

[43]  Zhihua Xie,et al.  Improving the robustness of the control volume finite element method with application to multiphase porous media flow , 2017 .

[44]  Christopher C. Pain,et al.  Non‐intrusive reduced‐order modeling for multiphase porous media flows using Smolyak sparse grids , 2017 .

[45]  D. Pavlidis,et al.  Improving the convergence behaviour of a fixed‐point‐iteration solver for multiphase flow in porous media , 2017 .

[46]  Ionel M. Navon,et al.  A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications , 2017 .

[47]  Min Chen,et al.  A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting , 2017, J. Comput. Phys..

[48]  Christopher C. Pain,et al.  Towards non-intrusive reduced order 3D free surface flow modelling , 2017, Ocean Engineering.

[49]  Ionel M. Navon,et al.  An efficient goal‐based reduced order model approach for targeted adaptive observations , 2017 .

[50]  Ionel M. Navon,et al.  Non‐intrusive reduced order modelling with least squares fitting on a sparse grid , 2017 .

[51]  James R. Percival,et al.  A force‐balanced control volume finite element method for multi‐phase porous media flow modelling , 2017 .

[52]  C. Pain,et al.  Model identification of reduced order fluid dynamics systems using deep learning , 2017, International Journal for Numerical Methods in Fluids.